On complete integral closedness of the $p$-adic completion of absolute integral closure

Abstract

Fix a prime $p$ and let $(R,\mathfrak{m})$ be a Noetherian complete local domain of mixed characteristic $(0,p)$ with fraction field $K$. Let $R^{+}$ denote the absolute integral closure of $R$, which is the integral closure of $R$ in an algebraic closure $\overline{K}$ of $K$. The first author has shown that $\widehat{R^{+}}$, the $p$-adic completion of $R^{+}$, is an integral domain. In this paper, we prove that $\widehat{R^{+}}$ is completely integrally closed in $\widehat{R^{+}}{\otimes }_{R^{+}}\overline{K}$, but $\widehat{R^{+}}$ is not completely integrally closed in its own fraction field when $\dim (R)\ge 2$.

15 pages, minor updates, final version